Tuesday, 17 January 2012

Why don't more mathematicians see the potential of economics

The question is, how did economics change its attitude to mathematics
in the forty years between Håvelmo’s The Probability Approach inEconometrics and his Nobel Prize in 1989, when he was pessimistic about
the impact the development of econometrics had had on the practice of
economics. Coinciding with Håvelmo’s pessimism, many economists were
reacting strongly against the ‘mathematisation’ of economics, evidenced
by the fact that before 1925, only around 5% of economics research
papers were based on mathematics, but by 1944, the year of Havelmo
and von Neumann-Morgenstern’s contributions, this had quintupled to
25%1.
While the proportion of economics papers being based on maths has not
continued this trajectory, the inﬂuence of mathematical economics has and the
person most closely associated with this change in economic practice was Paul
Samuelson.

Samuelson is widely regarded as the most inﬂuential economist to come out of
the United States and is possibly the most inﬂuential post-war economist in the
world. He was the ﬁrst U.S. citizen to be awarded the Nobel Prize in Economics in
1970 because “more than any other contemporary economist, he has contributed
to raising the general analytical and methodological level in economic
science”2.
He studied at the University of Chicago and then Harvard, were he obtained his
doctorate in 1941. In 1940 he was appointed to the economics department of M.I.T.,
in the ﬁnal years of the war he worked in Wiener’s group looking at gun control
problems3,
where he would remain for the rest of his life. Samuelson would comment that “I
was vaccinated early to understand that economics and physics could share the
same formal mathematical theorems”.

In 1947 Samuelson published Foundations of Economic Analysis, which laid
out the mathematics Samuelson felt was needed to understand economics. It is
said that von Neumann was invited to write a review Foundations in
1947 declined because “one would think the book about contemporary
with Newton”. Von Neumann, like many mathematicians who looked at
economics, believed economics needed better maths than it was being
oﬀered4.
In 1948 Samuelson published the ﬁrst edition of his most famous work,
Economics: An Introductory Analysis, one of the most inﬂuential textbooks on
economics ever published, it has run into nineteen editions and sold over four million copies.

There appears to be a contradiction, Håvelmo seems to think his introduction
of mathematics into economics was a failure, while Samuelson’s status seems to
suggest mathematics came to dominate economics. In the face of contradiction,
science should look for distinction.

I think the clue is in Samuelson’s attachment to “formal mathematical
theorems”, and that his conception of mathematics was very diﬀerent from that of
the earlier generation of mathematicians that included everyone from Newton and
Poincaré to von Neumann, Wiener and Kolmogorov.

A potted history of the philosophy of mathematics is that the numerologist
Plato came up with the Theory of Forms and then Euclid produced The Elements
which was supposed to capture the indubitability, the certainty, and immutability,
the permanence, of mathematics on the basis that mathematical objects where
Real representations of Forms. This was used by St Augustine of Hippo as
evidence for the indubitability and immutability of God, embedding into western
European culture the indubitability and immutability of mathematics. The
identiﬁcation of non-Euclidean geometries in the nineteenth century destroyed this
ediﬁce and the reaction was the attempt to lay the Foundations of Mathematics,
not on the basis of geometry but on the logic of the natural numbers.
Frege’s logicist attempt collapsed with Russell’s paradox and attention
turned to Hilbert’s formalismto provide a non-Platonic foundation for
mathematics. The key idea behind Formalism is that, unlike Platonic
Realism, mathematical objects have no meaning outside mathematics, the
discipline is a game played with symbols that have no relevance to human
experience.

The Platonist, Kurt Gödel, according to von Neumann, has “shown that
Hilbert’s program is essentially hopeless” and

The very concept of “absolute” mathematical rigour is not immutable.
The variability of the concept of rigour shows that something else
besides mathematical abstraction must enter into the makeup of
mathematics5

Mathematics split into two broad streams. Applied mathematics,
practised by the likes of von Neumann and Turing, responded
by focussing on real-world ‘special cases’, such as modelling the brain6.
Pure mathematics took the opposite approach, emphasising the generalisation of
special cases, as practised by Bourbaki and Hilbert’s heirs.

Formalism began to dominate mathematics in the 1940s-1950s. Mathematics
was about ‘rigorous’, whatever that means, deduction from axioms and deﬁnitions
to theorems. Explanatory, natural, language and, possibly worse, pictures, were to be
removed from mathematics. The “new math” program of the 1960s was a
consequence of this Formalist-Bourbaki dominance of mathematics.

It is diﬃcult to give a deﬁnitive explanation for why Formalism became
dominant, but it is often associated with the emergence of logical–positivism, a
somewhat incoherent synthesis of Mach’s desire to base science only on
phenomena (which rejected the atom), mathematical deduction and Comte’s
views on the unity of the physical and social sciences. Logical-positivism
dominated western science after the Second World War, spreading out from its
heart in central European physics, carried by refugees from Nazism.

The consequences of Formalism were felt most keenly in physics. Richard
Feynman, the physicists’ favourite physicist, hated its abandonment of relevance.
Murray Gell-Mann, another Noble Laureate physicist, commented in 1992 that
the Formalist-Bourbaki era seemed to be over

abstract mathematics reached out in so many directions and
became so seemingly abstruse that it appeared to have left physics
far behind, so that among all the new structures being explored by
mathematicians, the fraction that would even be of any interest
to science would be so small as not to make it worth the time of
a scientist to study them.

But all that has changed in the last decade or two. It has turned
out that the apparent divergence of pure mathematics from science
was partly an illusion produced by obscurantist, ultra-rigorous
language used by mathematicians, especially those of a Bourbaki
persuasion, and their reluctance to write up non–trivial examples
in explicit detail. When demystiﬁed, large chunks of modern mathematics
turn out to be connected with physics and other sciences, and
these chunks are mostly in or near the most prestigious parts
of mathematics, such as diﬀerential topology, where geometry,
algebra and analysis come together. Pure mathematics and science are ﬁnally being reunited and mercifully, the Bourbaki plague is
dying out.7

Economics has always doubted its credentials. Laplace saw the physical
sciences resting on calculus, while the social sciences would rest on
probability8,
but classical economists, like Walras, Jevons and Menger, wanted their emerging
discipline economics to have the same status as Newton’s physics, and so
mimicked physics. Samuelson was looking to do essentially the same thing,
economics would be indubitable and immutable if it looked like Formalist
mathematics, and in this respect he has been successful, the status of economics has grown faster than the growth of maths in economics. However, while the
general status of economics has exploded, its usefulness to most users of
economics, such as those in the ﬁnancial markets, has collapsed. Trading
ﬂoors are recruiting engineers and physicists, who always looked for the
relevance of mathematics, in preference to economists (or post-graduate
mathematicians).

My answer to the question “why don’t more economists see the potential of
mathematics” is both simple and complex. Economists have, in the main, been
looking at a peculiar manifestation of mathematics - Formalist-Bourbaki
mathematics - a type of mathematics that emerged in the 1920s in response to an intellectual
crisis in the Foundations of Mathematics. Economists have either embraced it, as
Samuelson did, or were repulsed by it, as Friedman was.

10 comments:

Thank you for the great discussion. While there is a great role for mathematics in economics, it does not seem to be comparable to physics, if only for the fact that economics ends up with people in the mix, which is an unpredictable and irrational element.

Perhaps rather than rely on existing genres of mathematics it is time for economists to develop their own?

Tim, I commented on your previous post. Here I note that mathematicians have to be paid by someone, and most people have wanted a short-term profit, so there has been no obvious incentive to apply, for example, Keynes' mathematics. What was used was quite good enough: we have had a form of arms-race in algorithms, where what matters is the race, not the 'validity' of the arms.

I was struck that your view of some of the 'Greats' is different from my own. Sometimes (as for Keynes) what the mainstream makes of a body of work is quite different from what they intended. Rather than economists needing more mathematics, I think it more important for them to understand the existing mathematics (and Greats).

Thirty years ago I had a brief chat with the mathematician Karen Uhlenbeck about Bourbaki, who praised Bourbaki for bringing order to mathematics, much as Dewey brings order to the books in any number of libraries. Her point was that mathematicians wished Bourbaki were not necessary, given the abstraction required; Bourbaki is something of a necessary evil.

The introduction of axiomatic methods into economics, on the other hand, is to be disparaged for two reasons. First, it attempts to substitute deductive for inductive reasoning; second, it distracts economists from examining techniques from applied, rather than pure, mathematics. The elegance of sigma algebras used to model efficient financial markets is a red herring given observed autocorrelative behaviors of markets. I feel a sense of schadenfreude watching option traders take revenge on pure mathematics by calling vega a Greek letter. All this abstraction balanced on the Black Scholes partial differential equation, a parabolic equation: why isn't anyone seeing if the other two classes of equations, the elliptic and the hyperbolic, may have useful application in modeling observed market behavior?

As to why Formalism was so popular around 1940, Russell's Paradox was a rather nasty blow to mathematics, along with other controversies like the Axiom of Choice. The trouble is that none of this has anything to do with mathematical modeling in the sciences.

The reason economists can't predict probably isn't that their reasoning is not robust enough (so that the solution might be to use axiomatic maths), but rather that there is too much to consider. (Too many industries, too much history, too many personalities.)

Thank you for this post. It provided a great context to the role of Mathematics in this particular field of study, especially with the intertwining historical facts you've exposed here.

However, to answer your question - coming from the 'Commerce' flip-side of your argument, I can't help but agree that mathematics is a core part of economics / finance. The mathematicians of the world need open their eyes to this..

The domination of mathematical economics by strict Bourbakism has been passe for for some time. This is made clear in Roy Weintraub's _How Economics Became a Mathematical Science_ from a decade ago, although he is not fully clear on what has succeeded it, which includes reliance on more inductive methods including computational ones, among other things.

not sure I agree with Gell-Mann's comment. two examples that come to mind are Thurston's work (both father and son, actually). A lot of ideas coming out of eg hyperbolic metric spaces (incl "outer space") don't, I think, relate to physics or science. Same with eg classification of tilings of the plane / 3-space. Homotopy probability; teichmueller theory; bordered floer homology; wheeling; seifert surfaces; commutative algebra; ---- just a couple things I've recently read about, talked about, or seen talks about which appear to come only from formal thought. How this would be applied (and probably never will be) is anyone's guess.

For a pure mathematician giving non-trivial examples and history, I recently read Steve kleiman's article in "fundamental algebraic geometry" called "history of the picard scheme". Best of both worlds: understands all the abstraction and